Automorphic Numbers with 10 Digits

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Theorem

The only $10$-digit automorphic numbers are:

$1 \, 787 \, 109 \, 376$
$8 \, 212 \, 890 \, 625$


Proof

We have:

\(\displaystyle 1 \, 787 \, 109 \, 376^2\) \(=\) \(\displaystyle \enspace 3 \, 193 \, 759 \, 92 \mathbf {1 \, 787 \, 109 \, 376}\)
\(\displaystyle 8 \, 212 \, 890 \, 625^2\) \(=\) \(\displaystyle 67 \, 451 \, 572 \, 41 \mathbf {8 \, 212 \, 890 \, 625}\)

thus demonstrating they are automorphic.

By Automorphic Numbers in Base 10, there are no others.

$\blacksquare$


Sources